The Brunn–Minkowski–Firey Theory II

Abstract

During the past two decades the notion of affine surface area (from affine differential geometry) and the isoperimetric inequalities related to it, have attracted increased interest. There are a number of reasons for this. First, there are new applications (see e.g. the survey of Gruber [12]). Then, there are the recently discovered extensions of affine surface area to arbitrary convex hypersurfaces (see e.g. Leichtwei? [15 18], Schu tt 6 Werner [37], Schu tt [36], Werner [38], and also [22]). These extensions have lead to recent verifications of the conjectured upper-semicontinuity and valuation property of classical as well as extended affine surface area (see [22,23,36]). Also, it has come to be recognized (see, e.g., [19,28,29]) that various isoperimetric inequalities involving affine surface area are very closely related to a variety of other important affine isoperimetric inequalities (e.g., the curvature image inequality, the Blaschke Santalo inequality, and Petty's geominimal surface area inequality).Geominimal surface area was introduced by Petty [30] more than two decades ago. As Petty stated, this concept serves as a bridge connecting affine differential geometry, relative differential geometry, and Minkowskian geometry. Both affine surface area and geominimal surface area are unimodular affine invariant functionals of convex hypersurfaces. Isoperimetric inequalities involving geominimal surface area are not only closely related to many isoperimetric inequalities involving affine surface area, but in fact, clarify the equality conditions of many of these inequalities.One of the aims of this article is to demonstrate that there are natural extensions of affine and geominimal surface areas in the Brunn Minkowski Firey theory. Surprisingly, it turns out that there are extensions of all of the known inequalities involving affine and geominimal surface areas to the new p-affine and p-geominimal surface areas. As will be article no. 0022 244

On successive radii of p-sums of convex bodies

“…Finally we consider the equality case in (17). If i = n, let S n be the n-dimensional simplex, embedded in R n+1 , lying in the hyperplane x = (x 1 , .…”

“…In [16,17] Lutwak studied p-sums of convex bodies systematically, and developed a theory nowadays known as Brunn-Minkowski-Firey theory. In the last years many important developments of this theory have come out; we mention e.g.…”

L p -Dual geominimal surface area

“…Obviously, if p = 1, G p (K) is just the geominimal surface area G(K). Further, Lutwak [3] proved the following result for the L p -geominimal surface area.…”

“…According to the L p -mixed volume, Lutwak [3] introduced the notion of L p -geominimal surface area. For K ∈ K n o , p ≥ 1, the L p -geominimal surface area, G p (K), of K is defined by…”

“…Associated with the L p -harmonic radial combination of star bodies, Lutwak [3] introduced the notion of L p -dual mixed volume as follows:…”